Multistep Brownian dynamics: application to short wormlike chains

Brownian dynamics simulations of short wormlike chains are carried out using the method of Ermak and McCammon [(1978) J. Chem. Phys. 69 , 1352–1360]. Following Hagerman and Zimm [(1981) Biopolymers 20 , 1481–1502], the wormlike chain is modeled as a string of beads. In each simulation, the dynamic evolution of an ensemble of 100 randomly generated chains is calculated for a period of from 3 to 200 ns. Two different “experiments,” fluorescence depolarization and dynamic light scattering, were performed in these simulations. Since we are primarily interested in the bending motions and not the torsional motions in this work, we have placed the transition moments along the local symmetry axis of the wormlike chain in the fluorescence depolarization “experiment.” As predicted by the Barkley and Zimm theory [(1979) J. Chem. Phys. 70 , 2991–3008], a considerable amount of rapid bending motion was detected by fluorescence depolarization, though not as much as predicted by theory. We conclude that these differences are primarily due to differences between the model used in the theory and the simulations. The light-scattering experiment was found to be insensitive to internal motion in the low scattering angle limit.     

Light scattering from wormlike chains with excluded volume effects

This paper reports a calculation of the angular dependence of light scattering from wormlike chains with excluded volume effects. The Daniels distribution function, modified for excluded volume effects, is used to compute averages for scattering elements separated by contour lengths which are long with respect to the persistence length of the chain. An expansion in terms of exactly known moments of the distribution for the wormlike coil without excluded volume is used for short contour lengths. The results are applied to scattering from calf thymus (M = 18.1 × 10⁶) and T7 (M = 25.4 × 10⁶) DNA. It is demonstrated that the same values of excluded volume parameter (ε = 0.11) and statistical segment length (1/λ′ = 900 Å) which explain the sedimentation and viscosity behavior of DNA also account satisfactorily for the scattering behavior. Molecular weights and root-mean-square radii estimated by extrapolation from scattering data obtained in the angular region from 10° to 25° will be 5–10% too large for DNA of molecular weight 20 × 10⁶–30 × 10⁶.     

Light-scattering and depolarization in biopolymers: Distinction between flexible chain models

The general formulas describing the low-angular dependence of polarized (V_v) and depolarized (H_v) light-scattering intensity by a chain macromolecule of arbitrary form with anisotropic polarizability are derived. It is shown that the value dH_v/dtH_vo (where t = [〈R²〉S/L]², S being the scattering vector) is an indication of the chain-flexibility mechanism. This permits one to distinguish between the models of a wormlike chain, regular zigzag, or statistical zigzag. The results of numerical calculations useful for the interpretation of experimental data are presented.     

Time-resolved fluorescence polarization anisotropy of short restriction fragments: the friction factor for rotation of DNA about its symmetry axis

The time-resolved fluorescence polarization anisotropy (FPA) of ethidium dye intercalated in 43 and 69 base pair (bp) restriction fragments is measured, and the friction factor per bp for rotation of DNA about its symmetry axis is determined. The same value of the hydrodynamic radius, a = 12.0 ± 0.6 Å, is obtained for both the 43- and 69-bp fragments, but only when (1) the twisting correlation functions appropriate for such short filaments are used: (2) the correct amplitude is employed for the uniform tumbling mode decays: and (3) the data analysis is restricted to times after the internal bending modes have died away leaving just reduced amplitudes of the exponentially decaying uniform tumbling modes. The present value of the hydrodynamic radius is significantly larger than that implied by the cross-sectional area perpendicular to the symmetry axis. This strongly suggests that a significant fraction of water in the major and minor groves is rotating more or less rigidly with the DNA on this time scale. The correct expression for the amplitude D_n(∞) of the uniform mode decay of the tumbling correlation function, including the average over all sites to which the dye could bind, is derived in the appendix. The present theory for D_n(∞) is compared with that of Barkley and Zimm [(1979) J. Chem. Phys. 70 , 2991–3007], and with recent Brownian simulations of discrete wormlike chains by Allison and co-workers [S. A. Allison and J. A. McCammon (1984) Biopolymers 23 , 363–375; S. A. Allison (1986) Macromolecules 19 , 118–124].     

Very low-angle light scattering. A characterization method for high-molecular-weight DNA

A very low-angle light-scattering photometer is described with respect to optical features, scattering cell, correction factors, and absolute calibration in the angular range 2°–35°. An improved microfiltration apparatus was used to obtain essentially dust-free aqueous solutions for very low-angle light scattering. The instrument was calibrated with silicotungstic acid, an absolute molecular-weight standard, and the calibration was confirmed with the use of several secondary standards. Very low-angle light-scattering measurements were made to determine the weight-average molecular weight M?_r and z-average radius of gyration R_g,z of a commerical preparation of calf-thymus DNA. Microfiltration of the solutions allowed measurements down to 6°. The value M?_r = 20.0 × 10⁶ obtained by extrapolating 6°–9° data to 0° is more than three times that from 30°–75° data (6.38 × 10⁶) but ～20% smaller than that from 10–35° data (23.7 × 10⁶). The experimental errors in M?_r and R_g,z are estimated to be ±8% and ±14%, respectively. Combined 6°–75° data from two photometers fit well a theoretical scattering curve for a model wormlike coil of the same M?_r as the DNA sample.     

Photon correlation spectroscopy, total intensity light scattering with laser radiation, and hydrodynamic studies of a well fractionated DNA sample.

A Malvern laser light-scattering instrument has been modified for use at scattering angles down to 5° and both total intensity and quasi-elastic scattering experiments. A sample of sheared, length-fractionated calf-thymus DNA was characterized by sedimentation, viscosity and electron microscopy. Quasi-elastic scattering and absolute intensity determinations were performed with the laser instrument and intensity determinations only with a Fica conventional light-scattering photometer. The total intensity experiments gave M?_w = (3.75 ± 0.15) × 10⁶ and 〈R²〉^1/2_z = (206.9 ± 10.3) nm which yielded a value for the persistence length, allowing for polydispersity, of 66 ± 6nm. The quasi-elastic experiments at scattering angles below 20° gave D⁰_{20, w} = (2.23 ± 0.06) × 10^?8 cm²/sec which combined with S⁰_{20, w} = 15.6 in the Svedberg equation gave M?_w = (3.73 ± 0.18) × 10⁶. In addition, from the higher angle data we extracted a value of the longest intramolecular relaxation time, τ1 of 17.5 msec. This is not in particularly good agreement with τ1 predicted by the Zimm–Rouse theory using our other experimental parameters. The disagreement may be due to the restricted applicability of the Zimm–Rouse spring-bead model as a quantitative representation of DNA molecules. Alternatively, it may be due to present difficulties in the unambiguous interpretation of molecular motions from the experimental autocorrelation functions.     

Light scattering study of natural DNAs over a wide range of molecular weights: Evidence for compaction of the large molecules

This work reports light scattering measurements on DNA in aqueous solutions (100 mM NaCl, 1 mM EDTA and 10 mM Tris–HCl buffer, pH 7.8) over a wide range of molecular weights (10²–10⁵ base pairs) and shows that, in the above standard solvent, shorter chains (<10⁴ base pairs) behave as a “wormlike chain” and their diffusion coefficients as obtained by dynamic light scattering measurements, confirm the prediction of standard wormlike model, whilst longer chains (>10⁴ base pairs) behave in a different manner. Dynamic and static light scattering and SEM analysis indicate that DNA molecules 10⁵ base pairs long, condense into compact structures in our solvent conditions. Calculations done using a wormlike model are also presented and discussed in comparison both to our experimental data and to other data reported in the literature.     

Dynamic light scattering studies of internal motions in DNA. II. Clean viral DNAs

Internal Brownian motions of clean ?29 and λ-DNAs have been studied using photon-correlation techniques at both visible (λ₀ = 632.8 nm) and uv (λ₀ = 363.8 nm) wavelengths. The present dynamic light scattering data, which extend to K² = 19 × 10¹⁰ cm^?2, can in every case be satisfactorily simulated by a Rouse-Zimm model polymer with an appropriate choice of the three model parameters. The effects of pH, salt concentration, single-strand breaks, and molecular weight on those model parameters have also been investigated. Intact clean DNAs exhibit surprisingly little variation with pH from 7.85 to 10.25, with salt concentration from 0.01 NaCl to 5.4M NH₄Cl, or with molecular weight or GC content. The single-strand breaks have no effect at pH 9.46, but produce dramatic changes in the model parameters at pH 10.0 and 10.25, indicating the introduction of titratable joints at those pHs. The failure of either single-strand breaks or a large change in GC content to alter the model parameters in the neutral pH range is a strong indication that local denaturation is not required for those flexions and torsions that dominate the relaxation of fluctuations in the scattered light. The Langevin relaxation time for the slowest internal mode of a particular Rouse-Zimm model derived from the dynamic light scattering data is compared with pertinent literature data extrapolated to the same molecular weight. The present algorithm for determining model parameters from the light-scattering D_app vs K² curve actually yields a Langevin time in fairly good agreement with the literature value. For unknown reasons the light-scattering D₀ values generally exceed those obtained from the molecular weight and sedimentation coefficient by about 20%.     

Brownian motion of the wormlike chain and segmental diffusion of DNA

The dynamics of the wormlike chain model for a polymer in solution is investigated in the case of free torsional and no longitudinal variations. A Langevin equation is derived and solved for circularly closed chains, neglecting hydrodynamic interactions. The local diffusion behavior of particular segments is described, and it is found that the mean-square displacements are proportional to t^3/4 at short times. Also, the equilibrium correlation functions for the closed chain are derived from the dynamic model in both the discrete and wormlike cases.     

Rotational effects in quasielastic laser light scattering from T-even bacteriophage.

We have applied a theory of dynamic light scattering from large anisotropic particles, developed by Aragón and Pecora [J. Chem. Phys. 66 , 2506–2516 (1977)] to calculate the scattering expected from T-even phage models. The results indicate that the off-center rotation of the massive virus head with respect to the center of frictional resistance introduces significant rotational contributions to the light-scattering time autocorrelation function. The effect is particularly important when the fibers of the phage are extended. Reanalysis of previously published data [J. B. Welch III and V. A. Bloomfield, Biopolymers 17 , 2001–2014 (1978)], taking into account rotational corrections, confirms the equality of molecular weights of the slow- and fast-sedimenting forms of T2L bacteriophage.     

Viscometry and sedimentation equilibrium of partially hydrolyzed hyaluronate: Comparison with theoretical models of wormlike chains

Fractionated samples of sodium hyaluronate of low molecular weight were used to calibrate the carbazole method for glucuronyl analsis and to determine the density increment (based on dry weight) of 0.444 (±0.003) mL/g in water and 0.386 (±0.003) mL/g for samples dialyzed against 0.2M NaCl. Weight-average molecular weights obtained by high-speed sedimentation equilibrium were used to calibrate the limiting viscosity number [η] in 0.2M NaCl, which gave [η]/M_w = 0.0028 (±0.0002) mL/g, valid to M_w = 0.0028 (±0.0002) mL/g, valid to M_w = 10⁵. Experimental data from this work and the literature, including viscosity and light- and small-angle x-ray scattering measurements, were compared to theoretical chain models of the Kratky-Porod (KP) wormlike and the helical wormlike (HW) chain, as treated by Yamakawa and collaborators. Although either model could be fitted to experimental data about equally well with consistent parameters, provided those for the HW chain were of weakly helical nature, calculation of the unperturbed meansquare end-to-end distance as a function of chain length from a conformational model favored the KP chain alternative. The parameters that provide the best fit to experimental data for the KP wormlike model are a persistence length of 4.5–5 nm and a diameter of 1.1 nm. The latter is resonable for a hydrated hydrodynamic cylinder in view of the approximate unhydrated value of 0.7 nm estimated from the density increment.     

Dynamic light scattering from thin rigid rods: Anisotropy of translational diffusion of tobacco mosaic virus

An Exact theoretical expression for the apparent diffusion coefficient D_app(K) of a thin rigid rod with arbitrary anisotropy of its translational diffusion diffusion coefficient is derived from the first cumulant of its dynamic structure factor. D_app(K) is predicted to reach a limiting plateau value at extermely large values of KL, where K is the scattering vector and L the rod length. Howerver, that limiting plateau value is approached only very slowly along a quasi-plateau with a very gradual slope. Dynamic light-scattering studies have been performed on tobacco mosaic virus from K² = (0.4–20) × 10¹⁰ cm^?2 using 632-8-nm laser radiation. The present data yield D₀ = (4.19 ± 0.10) × 10^?8 cm²/s (corrected to 20,w conditions) and, with literature data to establish L = 2980 Å and the rotational diffusion coefficient D_R = 318s^?1, yield also Δ ≡ D_∥ ? D_⊥ = (1.79 ± 0.38) × 10^?8 cm²/s. The experimental data closely follow the curve of D_app(K) vs K² calcuated for these parameters. The present value of D₀ substantially exceeds all previous dynamic light-scattering values, but is in good aggreement with previous sedimentation data, which were confirmed for the presemt sample. The anisotropy ratio Δ/D₀ = 0.43 ± 0.09 is in accord with theoretical predictions based on the modified Kirkwood algorithm, despite the fact the D₀ lies significantly below its corresponding theoretical value. The present data largely predlude the possibility that both D₀ and Δ/D₀ could simultaneously match their theoretical predictions. We present a detailed comparison of the experimental data with the calculations of Tirado and Garcia de la Torre based on the modified Kirkwood algorithm and with the Broersma formulas.     

Physicochemical studies on xylinan (acetan). I. Characterization by G permeation chromatography on sepharose Cl-2B coupled with static light scattering and viscometry

Laboratory-made samples of the polysaccharide xylinan (acetan) were fractionated on Sepharose Cl-2B using 0.1M NaCl as eluant. The weight average molar masses and intrinsic viscosities were estimated in the fractions by multiangle laser light scattering (off-line) and capillary viscometry, respectively. The Mark-Houwink-Sakurada plot was found to be indicative of semiflexible coils (a = 0.90). The angular dependence of scattered light was interpreted by fitting with theoretically calculated “Master Curves” in terms of a wormlike chain model. The ambiguity of the interpretation of scattering curves owing to the overlapping effects of chain stiffness and polydispersity is discussed in detail. The experimental data is found to be consistent with a persistence length of L_p = 100 nm. The main proportion consists of double-stranded chains (consistent with a robust double-helix), but single- and multistranded chains also are present. Our results suggest a fractionation according to the contour length rather than the molar mass. © 1996 John Wiley & Sons, Inc.     

Effect of molecular weight distribution on the solution properties of sodium hyaluronate in 0.2M NaCl solution

Various molecular parameters, which characterize sodium hyaluronate in 0.2M NaCl solution, were obtained at 25°C by means of the static and dynamic light scattering and low shear viscometry over the molecular weight range of 5.94–627 × 10⁴. Molecular weight distribution was obtained by using the Laplace inversion method of the autocorrelation function of the scattered light intensity and by Yamakawa theory for the wormlike chain with the stiff chain parameters for sodium hyaluronate in 0.2M NaCl (persistence length, chain diameter, molar mass per unit contour length, and the excluded‐volume strength). The molecular weight distribution thus obtained reproduced the solution properties of sodium hyaluronate well. Especially, the intrinsic viscosity showed a good agreement over four orders of molecular weight with Yamakawa theory combined with the Barrett function. Sodium hyaluronate in 0.2M NaCl solution is well expressed by the wormlike chain model affected by the excluded‐volume effect with the persistence length of 4.2 nm. © 1999 John Wiley & Sons, Inc. Biopoly 50: 87–98, 1999     

Dilute solution of amylose in dimethylsuldoxide

Measurements of light scatting, sedimentation equilibrium, sedimentation velocity, and viscosity were carried out on fractions of native amylose in dimethylsulfoxide at 25°C. The data for statistical radius as a function of weight-average molecular weight M_w suggested a stiff nature of this biopolymer in the solvent studied when interpreted in terms of Kirste's recent calculations with a stiff chain model. The data for sedimentation coefficient were consistent with this suggestion, and when analyzed in terms of the theory Hearst and Stockmayer for wormlike chain, a value of 233 Ų was obtainedd for a/λ, where a is the length of a monomer unit projected on the chain axis and (2λ)^?1is the persistence length of the wormlike chain. The intrinsic viscosity data gave a high a value as 0.91 for the exoponent in the Houwink-Mark-Sakuarada equation, in Substantial agreement with Cowie's prenious work. We attempted to interpret these data by use of the Eizner-Ptitsyn equation for wormlike chains, with omission of the free-drainage term and introduction of the a/λ value obtained from sedimentation data. It was found that, except in the region of M_w above one million, the observed values were fitted well by the E-P theory with a = 1.4 Å and (2λ)^?1 = 87 Å. The disagreement in the high-molecular-weight region was tentatively attributed to excluded volume effect. The a value obtained suggests that the molecular conformation of amylose in dimethylsulfoxide is predominantly helical, in contrast to that of the same polymer in aqueous solutions of simple electrolyte. It was also found that a similar value of a was derived from our data for the second virial coefficient and partial specific volume if the molecule was assumed to be essentially rodlike.     

Magnetic birefringence study of the electrostatic and intrinsic persistence length of DNA

Magnetic birefringence experiments were performed on solutions of DNA of intermediate molecular weight at several concentrations (c_p) over a wide range of ionic strengths (of NaCl and MgCl₂). The specific Cotton-Mouton constant (CM/c_p) is found to be independent of c_p when contributions from c_p to the ionic strength (μ_eff) are taken into account according to the concept of counterion condensation. For μ_eff ? 10^?2M, CM/c_p is also independent of the ionic strength; the plateau value results in an acceptable value of the intrinsic persistence length, when a revised theoretical expression for the magnetic birefringence of wormlike chains is used, combined with new experimental data for the monomeric optical and magnetic anisotropy. For μ_eff < 10^?2M, CM/c_p strongly, or wealky, increases with decreasing μ_eff, depending on the valency of the counterion used (Na⁺ or Mg²⁺, respectively). This increase agrees quantitatively with the variation of the electrostatic persistence length as predicted by Odijk [(1977) J. Polym. Sci. Polym. Phys. Ed. 15 , 477–483], Odijk and Houwaart [(1978) J. Polym. Sci. Polym. Phys. Ed. 16 , 627–639], and by Skolnick and Fixman [(1977) Macromolecules 10 , 944–948]. A comparison with other experimental data seems to reveal the importance of excluded-volume effects, which are particularly pronounced in the low-salt regime.     

Dependence of laser light scattering of DNA on NaCl concentration

We show that the persistence length a of DNA, derived from total intensity laser light scattering of linear Col E₁ DNA and corrected for excluded-volume effects, varies from about 68 nm in 0.005M NaCl to about 40 nm in 0.2M NaCl, leveling off to a constant value (about 27 nm) at high NaCl (1–4M) concentration. These observations do not agree with current views on the effect of electrostatic charge and ionic conditions on DNA dimensions. The apparent diffusion constant D_app, determined from laser light scattering autocorrelation as a function of scattering vector q, at NaCl concentrations 0.005–4M, correctly yields the translational diffusion coefficient D_t at low values of q and scales with molecular dimensions rather than segment length at high values of q; thus, D_app/D_t yields a universal curve when plotted against q²R, where R_g is the radius of the gyration. The sedimentation coefficients s at 0.1 and 0.2M NaCl concentration closely agree with the well-tested empirical relations, and a combination of s, D_t, and the appropriate density increments yield correct molar masses over the whole salt concentration range. Approximate constancy of D_tR_g indicates limited draining in translational flow. We present some observations and thoughts on the regimes in which a dependence of the correlation decay times on q³ rather than q² applies. We conclude that quasielastic laser light scattering discloses little information about dynamics of internal motion of DNA chains.     

Light scattering and hydrodynamic properties of linear and circular bacteriophage lambda DNA

Low-angle light scattering, sedimentation velocity, and intrinsic viscosity measurements have been made on circular and linear forms of lambda (λ) bacteriophage DNA. Available equations, used to relate hydrodynamic parameters of both forms to the molecular weight, give relatively consistent values of particle weights which essentially agree with the light-scattering results. An average molecular weight of (34 ± 3) × 10⁶ for λ DNA was obtained in good agreement with literature values of (31–33) × 10⁶. The linear λ DNA has a larger root-mean-square radius than the circular molecule, when determined by light scattering, but the difference does not appear to be us large as expected from hydrodynamic data. The two forms also show significantly different angular distrbutions of scattered light intensities which agree only qualitatively with those derived from existing theory. The light-scattering results suggest that further experiments and modifications of available theories should be undertaken.     

Physicochemical studies on xylinan (acetan). III. Hydrodynamic characterization by analytical ultracentrifugation and dynamic light scattering

A laboratory-made sample of the polysaccharide xylinan (acetan) has been further characterized with respect to (i) purity, (ii) molar mass and polydispersity, and (iii) gross conformation by a combination of hydrodynamic measurements (sedimentation velocity and equilibrium analytical ultracentrifugation, viscometry, and dynamic light scattering) in aqueous NaCl (I = 0.10 mol·L⁻¹). Sedimentation velocity diagrams recorded using Schlieren optics revealed highly pure material sedimenting as a single boundary [s^o_20.w = 9.5 ± 0.7) S; k_s = (273 ± 112) mL/g]. The hypersharp nature of these boundaries is symptomatic of a polydisperse and highly nonideal (in the thermodynamic sense) system. Low speed sedimentation equilibrium in the analytical ultracentrifuge using Rayleigh interference optics and two different types of extrapolation procedure (involving point and whole-cell molar masses) gave a weight average molar mass M_w of (2.5 ± 0.5) × 10⁻⁶ g·mol⁻¹ and also a second virial coefficient, B = (2.8 ± 0.7) × 10⁻⁴ mL·mol·g⁻², both values in good agreement with those from light scattering-based procedures (Part II of this series). A dynamic Zimm plot from dynamic light scattering measurements gave a z-average translational diffusion coefficient D^o_20.w = (3.02 ± 0.05) × 10⁻⁸ cm²·s⁻¹ and the concentration-dependence parameter k_D = (370 ± 15) mL/g. Combination of s^o_20.w with D^o_20.w via the Svedberg equation gave another estimate for M_w of ≅ 2.4 × 10⁶ g/mol, again in good agreement. Both the Wales-van Holde ratio (k_s/[η]) ≅ 0.4 (with [η] = (760 ± 77) mL/g) and the ρ-parameter (ratio of the radius of gyration from static light scattering to the hydrodynamic radius from dynamic light scattering) as ρ > 2.0 all indicate an extended conformation for the macromolecules in solution. These findings, plus Rinde-type simulations of the sedimentation equilibrium data are all consistent with the interpretation in terms of a unimodal wormlike coil model performed earlier. © 1996 John Wiley & Sons, Inc.     

Dynamic light-scattering study of semiflexible polymers: collagen

Dynamic light-scattering measurements were carried out for collagen in acetate buffer (pH 4.8) extracted from lathyritic ratskin. The correlation functions were analyzed in terms of the semiflexibility of collagen molecules. The experimental Γ /K² vs K² relationship was compared with the theoretical one based on formulation including anisotropy in translational diffusion, chain flexibility, and the hydrodynamic interaction; Γ is the average decay rate and K is the magnitude of the momentum transfer vector. By using the model parameters evaluated from the Γ /K² vs K² relationship, a good agreement was obtained between profiles of theoretical and experimental correlation functions over the entire delay times. Detailed examinations of the dynamic light-scattering spectrum permitted us to conclude that a set of the contour length L of 300 nm and the Kuhn length γ^?1 of 340 nm are much more probable than other sets of L and γ^?1 that equally explain static quantities such as the radius of gyration. The results show that collagen molecules are well characterized by a wormlike chain model.